Subgroups — #22577674

Order 8768.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17984116	C₂	1	2	yes	maximal, complemented
2	C₂×D₁₀₉₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#17984094	C₂	1	1	yes	maximal, characteristic, complemented
4	#17984004	C₂	1	2	yes	maximal, complemented
5	#108×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#14555510	C₂²	1	1	yes	characteristic
7	C₂×D₅₄₈	C₂²	1	2	yes	—
8	#14555499	—	2	2	no	—
9	D₁₀₉₆	—	2	2	no	—
10	D₁₀₉₆	—	2	1	no	—
11	D₁₀₉₆	—	2	1	no	—
12	#14555494	C₂²	1	1	yes	characteristic
13	#14555493	—	2	1	no	—
14	C₈×D₁₃₇	—	2	1	no	—
15	M₁₆×C₁₃₇	C₂²	1	2	yes	—
16	C₂×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
17	C₂²×D₁₃₇	—	2	2	no	—
18	Dic₂₇₄	D₄	1	1	yes	characteristic
19	D₅₄₈	—	2	2	no	—
20	D₅₄₈	—	2	2	no	—
21	D₅₄₈	D₄	1	1	yes	characteristic
22	C₂²⋊D₁₃₇	—	2	1	no	—
23	C₄×D₁₃₇	—	2	1	no	—
24	#11764040	—	2	1	no	—
25	C₂×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
26	C₈×C₁₃₇	—	2	2	no	—
27	C₈×C₁₃₇	D₄	1	1	yes	characteristic
28	C₈×C₁₃₇	D₄	1	1	yes	characteristic
29	C₂²×C₁₃₇	#41	1	1	yes	characteristic
30	C₂×D₁₃₇	—	2	1	no	—
31	C₂×D₁₃₇	—	4	2	no	—
32	C₂×D₁₃₇	—	8	2	no	—
33	C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
34	C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
35	Dic₁₃₇	—	2	1	no	—
36	C₂×C₁₃₇	—	2	1	no	—
37	C₂×C₁₃₇	#121	1	1	yes	characteristic
38	D₁₃₇	—	4	1	no	—
39	D₁₃₇	—	8	2	no	—
40	C₁₃₇	#472	1	1	yes	characteristic, complemented, Sylow
41	#472	—	137	1	no	maximal, Sylow
42	AGL(1,ℤ/8)	—	137	2	no	—
43	C₂×D₈	—	137	1	no	—
44	#131	—	137	1	no	—
45	#108	D₁₃₇	1	1	yes	characteristic, complemented
46	#100	—	137	2	no	—
47	#41	—	137	1	no	—
48	C₂×D₄	—	137	2	no	—
49	SD₁₆	—	274	2	no	—
50	D₈	—	274	2	no	—
51	D₈	—	274	1	no	—
52	D₈	—	274	1	no	—
53	M₁₆	—	274	1	no	—
54	M₁₆	—	137	1	no	—
55	M₁₆	C₂×D₁₃₇	1	2	yes	—
56	C₂×C₈	—	274	1	no	—
57	C₂×C₈	C₂×D₁₃₇	1	1	yes	characteristic
58	C₂³	—	274	2	no	—
59	Q₈	—	137	1	no	—
60	D₄	—	274	1	no	—
61	D₄	—	274	2	no	—
62	D₄	—	274	2	no	—
63	D₄	—	137	1	no	—
64	C₂×C₄	—	274	1	no	—
65	C₂×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
66	C₈	—	2	2	no	—
67	C₈	—	274	1	no	—
68	C₈	D₅₄₈	1	1	yes	characteristic
69	C₈	D₅₄₈	1	1	yes	characteristic
70	C₂²	—	274	1	no	—
71	C₂²	—	548	2	no	—
72	C₂²	—	1096	2	no	—
73	C₂²	#14555511	1	1	yes	characteristic
74	C₄	—	274	1	no	—
75	C₄	D₄×D₁₃₇	1	1	yes	characteristic
76	C₄	C₂×D₅₄₈	1	1	yes	characteristic
77	C₂	—	2	1	no	—
78	C₂	—	548	1	no	—
79	C₂	—	1096	2	no	—
80	C₂	#17984063	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#17984063
commutator	C₂×C₄×C₁₃₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₃₇
Fitting	#108×C₁₃₇	C₂
socle	C₂×C₁₃₇	#121
hypercenter	#108	D₁₃₇
hypocenter	C₁₃₇	#472
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #108

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #17984116	C₂	semidirect
2. C₂×D₁₀₉₆	C₂	semidirect
3. #17984094	C₂	semidirect
4. #17984004	C₂	semidirect
5. #108×C₁₃₇	C₂	semidirect
40. C₁₃₇	#472	semidirect
45. #108	D₁₃₇	semidirect